Despite the broad occurrence of anomalous diffusion in mesoscopic and
biological systems, applications of concepts of ST to such systems are
still rare, and many open questions remain. This concerns both
anomalous models with Markovian and non-Markovian character. Existing
studies mainly focus on FRs. For example, a series of papers using
(non-Markovian) GLEs has confirmed the validity of the Crooks and the
Jarzynski FRs, as well as of transient and steady state FRs
[1]}, [2]}, [3]}, [4]}, [5]}. More
generally, the validity of (different) FRs for GLE-like dynamics has
been shown in Ref. [6]}. Notably, the
above-mentioned results in the framework of GLEs have been obtained
under the assumption that the noise correlation function and the
memory kernel are related (in fact, proportional) to each other by the
FDR of the second kind [7]}.
If
this restriction fails, such as in overdamped GLE models of
driven systems without FDR [8]}, [9]}, the conventional form of FRs may not be
obtained for thermodynamic observables. This problem is also
explored in very recent works modeling fluctuations of a Brownian
particle in an active bath, for which GLEs with two different kinds
of noises have been used, typically Gaussian white and coloured
noise [10]}. For the latter, representing the active bath,
FDR is broken, and deviations from conventional FRs have been
reported, arising in such models
[10]}, [12]}, [13]}, [14]}, [15]}, [16]}. Beyond GLE
models, forms different from conventional (steady state and
transient) FRs, dubbed anomalous fluctuation relations
[17]}, were also obtained for systems
with non-Gaussian noises [18]}, [19]}, [20]}, [21]}, [22]}, [23]}, [24]}, [25]}, in glassy systems
[26]}, [27]}, and in
continuous time random walks for certain exponents of the (power-law)
waiting time distribution [28]}. More
recently, studies of FRs and further concepts of ST have been extended
to other non-trivial systems of current interest, particularly
to active particles
[10]}, [30]}, [31]}, [32]}
and systems with time-delay
[33]}, [34]}, [35]}, [36]}.
We also mention recent studies of TURs in systems displaying anomalous
dynamics [37]} and time delay
[38]}. All these developments highlight the
ongoing strong interest in understanding the ST of systems beyond
standard Brownian motion. However, to the best of our knowledge, most
studies have focused on specific aspects (such as FR, TUR), while the
general framework of ST for anomalous processes seems still
underdeveloped.
